Mixed methods
\(ATE = \lambda^Y_{01} - \lambda^Y_{10}\)
We need to
Key insight:
That, with priors, is enough to update:
\[p(\lambda | D) = \frac{p(D | \lambda)p(\lambda)}{p(D)}\]
Data on exogenous variables and a key outcome for many cases
E.g., data on inequality (\(I\)) and democracy (\(D\)) for many cases
CausalQueries uses information wherever it finds itFor Bayesian approaches this mixing is not hard.
Critically though we maintain the assumption that cases for “in depth” analysis are chose at random—otherwise we have to account for selection processes.
What is the probability of seeing these two cases:
given parameters \(\lambda\):
The probability of 1 is:
\[p_{111}= \lambda^X_1 \times (\lambda^M_{01} + \lambda^M_{11}) \times (\lambda^Y_{01} +\lambda^Y_{11})\]
The probability of 2 is:
\[p_{1?1} = \lambda^X_1\times \left((\lambda^M_{01} + \lambda^M_{11}) \times (\lambda^Y_{01} +\lambda^Y_{11}) + (\lambda^M_{10} + \lambda^M_{00}) \times (\lambda^Y_{10} +\lambda^Y_{11}) \right)\]
So the probability of this data is just:
\[p(D|\lambda) = p_{111} * p{1?1}\]
Insight:
If we imagine possible parameter values we can figure out the likeihood of any data type – quantitative, qualitative or mixed.
That, with priors, is enough to update:
\[p(\lambda | D) = \frac{p(D | \lambda)}{p(D)}\]
Remember:
Remember
Suppose we go to the field and we learn that mass mobilization DID occur in Malawi
What can we conclude?
NOTHING YET!
CausalQueries